Rand Paul was right that most Americans pay far less in taxes into the Medicare program than they collect in benefits.

According to estimates researchers at the Urban Institute made in 2013, a single man earning the median income who turns 65 this year will pay about $70,000 in lifetime Medicare taxes and collect about $197,000 worth of lifetime benefits.

But his assertion that gradually raising the age of eligibility for the program is “the only way you fix Medicare” exaggerates the effects of such a policy.

Most money spent in the program is spent on older seniors, not those new to the program. Recent estimates from the Congressional Budget Office suggest that raising the eligibility age from 65 to 67 would save only $19 billion over 10 years, a tiny fraction of the program’s budget, about $500 billion this year alone.

Responding to a question about H-1B visas for highly skilled foreign workers, Senator Marco Rubio said he would require companies to advertise a job for 180 days domestically before it could be filled by a foreign worker with one of those visas. And he would require companies to prove that they would pay foreign workers more so wages for Americans would not be undercut.

But neither of those proposals is in a bill to increase H-1B visas that Mr. Rubio, along with other senators, sponsored this year.

That bill, introduced in January, does not add any new protections from the existing H-1B program, which does not have a requirement to recruit American workers first and has led to many Americans being displaced by foreign workers with those visas.

There are plenty of vast divides between the Democratic and Republican candidates for president. But none may be deeper than when it comes to the issue of guns.

Donald J. Trump made that crystal clear Wednesday night when he acknowledged that he sometimes carries a gun — with a permit — and believes that gun-free zones are an invitation to massacres.

“Gun-free zones. When you say that, that’s target practice for the sickos and the mentally ill,” Mr. Trump said. “I think gun-free zones are a catastrophe.”

That assertion drew no criticism from any of his competitors on the stage, sending the message that the Republican Party is at least somewhat unified behind that opinion.

That is strikingly different from the Democratic debate, where calls for new gun-safety measures were just as unified. Hillary Rodham Clinton, in particular, has urged more gun-control measures, and is telegraphing that she would make the issue a central part of a general election campaign.

If so, she is almost certain to find a Republican opponent who is fiercely on the other side.

And — if it’s Mr. Trump — maybe armed, too.

10:10 pm ETOct 28, 20152015-10-29T02:23:40+00:00

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Senator Marco Rubio, Donald J. Trump, Ben Carson and Carly Fiorina onstage during the Republican presidential debate on Wednesday at the University of Colorado in Boulder.Credit Jim Wilson/The New York Times

If there is one sure-fire applause line at every Republican debate, it is to attack the “mainstream” media. And the candidates did not disappoint Wednesday night.

Donald J. Trump got a big laugh when he challenged a question about his opposition to high-tech immigration visas, saying, “You people write this stuff.” Earlier he accused one of the debate moderators of asking “a not-very-nicely asked question.”

Others on the stage got easy applause from suggesting that CNBC was trying to pit the candidates against each other, rather than focusing on policy questions.

But it has been Marco Rubio who got in the most pointed critiques of the media.

When he was questioned about his personal financial difficulties — and whether he has the “maturity and wisdom” to be president — he responded by conflating the media and Democrats.

“You just listed a litany of discredited attacks by Democrats,” he said.

Later, though, he was even more direct. Responding to Mr. Trump’s accusation that “super PACs” are a danger to Democracy, Mr. Rubio said that “Democrats have the ultimate super PAC: It’s called the mainstream media.”

He got huge applause. And he kept going. He accused the media of being on Hillary Rodham Clinton’s side when she testified in Congress recently about the Benghazi attacks.

“It was the week she got exposed as a liar,” Mr. Rubio said, “but she has her super PAC helping her out, the American mainstream media.”

Carly Fiorina said that “92 percent of the jobs lost during Barack Obama’s first term belonged to women” as part of a discussion of gender differences in compensation.

The assertion is flawed. In January 2009, the month President Obama’s first term began, 66.9 million women were employed in the United States, according to data from the Bureau of Labor Statistics. In January 2013, when his first term ended, that number had risen slightly, to 67.1 million. Overall employment similarly rose slightly, from 142.15 million to 143.32 million, meaning the pattern of employment among women in Obama’s first term was similar to the pattern among men.

In other words, overall employment among both women and men rose during Mr. Obama’s first term, though very slightly. That said, the unemployment rate among both groups was higher at the end of his first term than the beginning, with the jobless rate among women rising to 7.8 percent in January 2013 from 7 percent in January 2009, reflecting a growing population

9:33 pm ETOct 28, 20152015-10-29T01:38:19+00:00

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John Kasich, left, Mike Huckabee, Jeb Bush, Marco Rubio, Donald J. Trump, Ben Carson, Carly Fiorina, Ted Cruz, Chris Christie and Rand Paul on Wednesday during the debate at the University of Colorado.Credit Jim Wilson/The New York Times

Carly Fiorina got a big boost from an earlier debate, and then her campaign faded a bit. Her response this time: hit hard on the theme of the big and powerful versus the small and the powerless.

It’s a fascinating argument for a woman who once was squarely in the former category. She served as the chief executive of Hewlett-Packard, one of the nation’s largest, richest companies. She and her company were definitely the big and powerful.

But in the wake of Mitt Romney’s struggles in the 2012 campaign, it’s clear that being perceived that way won’t work with voters. And worse, the country has shifted even more since then toward a populism that questions the benefits that go to the rich and powerful.

Mrs. Fiorina clearly gets that. In her remarks during the debate, she repeatedly talked about what she called “crony capitalism,” which she defined as big, powerful companies partnering with a big, powerful government to get ahead at the expense of others.

“This is how socialism starts,” she said. “Government causes a problem, and then government steps in to solve a problem.”

Will it work? Her Republican colleagues have so far resisted attacking her C.E.O. credentials. (With the exception of Donald J. Trump, who has in the past accused her of being a terrible leader of the company.) But the Democrats would not be so friendly if she were to get the nomination.

President Obama’s aides seized on Mr. Romney’s business background and — with the help of a leaked audio recording of comments that Mr. Romney made about the “47 percent” — helped turn that background into a political albatross.

Mrs. Fiorina is hoping to avoid the same fate if she becomes the Republican nominee.

Gov. Chris Christie of New Jersey proudly proclaimed that he is the “only person who’s put out a detailed plan on how to deal with entitlements.”

While he has made this knotty problem a focus of his campaign, he is not the only candidate who has put forth proposals on entitlements: Jeb Bush released a detailed plan on Tuesday to reform Medicare and Social Security to assure their solvency. And Mr. Christie’s detailed plan is not so detailed: it is all of three pages.

9:12 pm ETOct 28, 20152015-10-29T01:36:18+00:00

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Senator Marco Rubio responded to a question about his finances by saying he was “worried about the finances of everyday Americans.”Credit Jim Wilson/The New York Times

Gov. Chris Christie of New Jersey criticized the way the government has handled Social Security payments.Credit Jim Wilson/The New York Times

Gov. Chris Christie of New Jersey, in his tell-it-like-it-is campaign, says that the government lied to the American people and told them that their Social Security payments were in a surplus even as it stole the money and left them a “pile of IOUs.”

That is true — kind of.

Social Security taxes have generated far more money every year than was spent on benefits. Instead of putting that surplus into a bank account, the government has taken it and spent it on other programs, like the military.

The only time in recent years that hasn’t happened was the very end of the Clinton White House, when the surplus was so large that Social Security receipts did not have to be tapped and Vice President Al Gore promised to put it in a “lockbox.”

But that pile of IOUs is not a bunch of scrap paper. They are federal Treasury notes. And by law, the government must cash them for benefits when they are due. Yes, the cashing of those bonds will cause the deficit to grow, but the IOUs are good money.

People paid their money, they expect to have it. And if this government doesn’t pay it, then tell me what’s different between the government and Bernie Madoff, who sits in prison today for doing less than what the government has done to the people on Social Security and Medicare in this country.

They might have once been friends, but Jeb Bush and Marco Rubio are clearly feeling the pressure that a competitive bid for president creates.

Seeking to save his faltering campaign, Mr. Bush seized on criticism that Mr. Rubio is failing to do his job as a senator while campaigning for president. Mr. Bush called on Mr. Rubio to resign.

“Marco, when you signed up for this it was a six year term,” Mr. Bush said. “You should show up for work.”

The anger from Mr. Rubio — who once described Mr. Bush as his mentor — was clear. He accused Mr. Bush of saying he wanted to model his own campaign on the one ran in 2008 by John McCain, who also missed a large number of votes in the Senate.

The tension between the two men highlights not only the competition between two of Florida’s leading Republicans, but also the struggles of the party’s establishment figures in a campaign dominated by Donald J. Trump and Ben Carson.

Mr. Bush and Mr. Rubio have both been polling in the single digits, and attacks on Mr. Trump and Mr. Carson have failed to work, especially for Mr. Bush. In the meantime, the two men are battling for the position of the alternative to the outsiders.

That means using the prime-time audience Wednesday night to highlight the differences between them. Expect more such exchanges before the night is out.

In the debate’s first terse exchange, Donald J. Trump blamed Gov. John Kasich of Ohio for the financial crisis, citing his work for Lehman Brothers, which collapsed in 2008, sending the economy into a downward spiral.

While Mr. Kasich did work for the now-bankrupt Lehman Brothers, the truth is that he was hardly steering the doomed ship in New York. Mr. Trump claimed that he was on the board, which was not true.

Instead, Mr. Kasich worked out of a two-man office in Columbus, Ohio, on deals with other bankers. He has said that he shared no blame for the bank’s troubles, precipitated by risky mortgage investments.

Senator Marco Rubio of Florida took it on the chin Wednesday morning when The Fort Lauderdale Sun-Sentinel — a newspaper that had endorsed him for the Senate — said he should either do his job in the Senate or resign.

Mr. Rubio didn’t defend his record of absenteeism because he has missed a lot of votes, a lot more than the other senators running for president: Ted Cruz, Rand Paul, Lindsey Graham and Bernie Sanders.

Instead, he compared himself to other senators like John McCain and Bob Graham, who also missed a lot of votes during their runs for president.

One problem: Those senators never intended to give up their seats if they weren’t elected. Mr. Rubio has already said he will not run for re-election. He is biding time before his departure.

His position is more like Bob Dole than John McCain. And Mr. Dole did resign from the Senate to pursue his 1996 presidential run.

For the record, in 2015, Mr. Rubio has missed 99 votes, giving him an attendance percentage of 65.98. Recently, Mr. Rubio missed votes on cybersecurity, appropriations, defense authorization and the Iran resolution of disapproval.

8:50 pm ETOct 28, 20152015-10-29T00:50:35+00:00

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Carly Fiorina and Senator Ted Cruz.Credit Jim Wilson/The New York Times

The debate on Wednesday night might be the last chance for the struggling Republican candidates, like Mr. Kasich, to grab the spotlight, and he wasted no time in attacking his party’s front-runners in an effort to seize the focus of the camera and perhaps bolster his campaign.

In the opening moments of the CNBC debate, Mr. Kasich ripped into Ben Carson’s tax proposals and Donald J. Trump’s immigration plans — and he minced no words, accusing his party colleagues of engaging in “fantasy” and saying voters should not elect “somebody who doesn’t know how to do the job.”

“You don’t just make promises like this. Why don’t we just give a chicken in every pot?” Mr. Kasich bellowed, adding, “Folks we’ve got to wake up.”

The quick — and pointed — exchanges suggest that the friendly back-and-forth in the still-crowded Republican race are now firmly ended. And without making a quick mark in the debate, some of those candidate fear they may simply fade from view just as early voting begins.

But the leading candidates are not going to take his attacks lightly, as Mr. Trump showed. Attacked by Mr. Kasich on his proposal to deport 11 million illegal immigrants, Mr. Trump attacked Mr. Kasich as partly responsible for the collapse of the Lehman Brothers investment bank in 2008.

“He was on the board,” Mr. Trump charged, accusing Mr. Kasich of getting desperate in the race for president. “He was such a nice guy. Then his poll numbers tanked.”

But Marco, when you signed up for this, this was a six-year term and you should be showing up to work. I mean literally, the Senate, what is it like, a French workweek? You get, like, three days where you have to show up?

Gov. John Kasich of Ohio on Wednesday during the debate in Boulder, Colo.Credit Jim Wilson/The New York Times

Gov. John Kasich of Ohio said he was the chief architect of the balanced budget in the 1990s. Hmmm. While it is true that Mr. Kasich was at that time the chairman of the House Budget Committee, most economists say the surpluses of the 1990s were because of a combination of three factors: The “read my lips” tax increases and spending cuts of President George Bush in 1991; President Bill Clinton’s budget of 1993, which most likely cost Democrats control of Congress; and the gold rush economy of the dot-com bubble.

Newt Gingrich, Mr. Kasich and Mr. Clinton reached their balanced budget agreement in 1997, but little of it had actually kicked in when the government’s red ink disappeared and surpluses swelled.